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Every commutative ring is the directed colimit of its subrings that are finitely generated as $\mathbb{Z}$-algebras. The Hilbert Basis Theorem implies that these subrings are noetherian. Actually this method is used in EGA IV, §8.9 to generalize some theorems from noetherian schemes to more general schemes.

Should "of its subrings" be replaced by something like "of its finitely-generated subrings"? Surely you meant the compact elements of the lattice of subrings, not the entire lattice?
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Andrej BauerNov 12 '11 at 13:21

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Not that I am well placed to make such remarks, but I believe at least some would say that (then) 'which' should be 'that' (to signal the restrictive nature of the clause).
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quidNov 12 '11 at 14:45

Oh, it was a linguistic issue, not a mathematical one. You meant "directed colimit of those subrings which are finitely generated as $\mathbb{Z}$-algebras", not "directed colimits of all subrings, which by the way are finitely generated as $\mathbb{Z}$-algebras.
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Andrej BauerNov 12 '11 at 16:22